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  1. Define a new topology on the plane as follows. If p is a point on the plane, we say that U is a neighborhood of p if every straight line that contains p also contains an open line segment that is contained in U. Compare this topology with the usual topology on the plane (i.e., is it strictly weaker? strictly coarser? neither? are the two topologies equivalent?)

I am trying to define this topolgy.
I have $$\tau=\{U \subseteq X |\text{ if } (a,b)\cap p\in U, \text{ then } U \in N_{p}\}$$ where $N_{p}$ is the neighborhood system at p.
Is this correct? If so, am I writing it right? Something about it feels off.

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Show for any point p in an open ball has a line through p containing an open line segement segment l with p in l subset U.

From that show the fuzzy ball topology is much finer than the smooth topology for the plane.

Show that the boundary of a fuzzy ball may not be smooth, can have sharp points.

Generalize the fuzzy plane to n dimensions.

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  • $\begingroup$ So is the way I defined the topology correct? $\endgroup$ – Kim Oct 22 at 17:53
  • $\begingroup$ @Kim. No....... $\endgroup$ – William Elliot Oct 22 at 23:08

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